Nonparametric Regression for Spherical Data

Abstract

We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability, and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression. Such current methods are surely not widely employable since they have a parametric nature, and also require the same dimensionality for prediction and response spaces, along with nonrandom design. Our approach does not suffer these limitations. Real-data case studies and simulation experiments are used to illustrate the effectiveness of the method.

Nonparametric regression using needlet kernels for spherical data

“…As the sphere is topologically a compact two-point homogeneous manifold, some widely used schemes for the Euclidean space such as the neural networks [14] and support vector machines [32] are no more the most appropriate methods for tackling spherical data. Designing efficient and exclusive approaches to extract useful information from spherical data has been a recent focus in statistical learning [11,21,28,31].…”

“…A major problem is that the stereographic projection usually leads to a distorted theoretical analysis paradigm and a relatively sophisticate statistical behavior. Localization methods, such as the Nadaraya-Watson-like estimate [31], local polynomial estimate [3] and local linear estimate [21] are also alternate and interesting nonparametric approaches. Unfortunately, the manifold structure of the sphere is not well taken into account in these approaches.…”

Power Curve Estimation With Multivariate Environmental Factors for Inland and Offshore Wind Farms

“…The kernel function for D is chosen to be the von Mises kernel (Taylor 2008), because it is a circular variable that may cause trouble in numerical computation; for more comprehensive discussion regarding the handling of circular variables, please refer to (Marzio et al 2012;Marzio et al 2013;Marzio et al 2014). The von Mises kernel function can characterize the directionality of a circular variable and takes the form…”

“…The specific approaches employed in these studies differed: Nielsen et al (2002) used a local polynomial regression; Sanchez (2006) presented a dynamic combination of several prediction models based on time-varying coefficients and a recursive solution procedure; Pinson et al (2008) used a total least squares criterion (i.e., orthogonal distance least squares), together with a Huber M-estimator, to achieve a certain degree of robustness. Kernel methods are among the sophisticated approaches that are used to model wind direction predictions (Marzio, Panzera and Taylor 2012;Marzio, Panzera and Taylor 2013;Marzio, Panzera and Taylor 2014) and model the power to wind speed/direction relationship through Jeon and Taylor (2012). Not only does Jeon and Taylor (2012) consider both wind speed and wind direction, it also produces a density estimation that can be used to account for uncertainty in wind power prediction.…”

Optimal designs for regression with spherical data

“…Several methods for estimating a spherical regression function nonparameterically have been proposed in the literature. Di Marzio et al (2009Marzio et al ( , 2014 investigate kernel type methods, while spherical splines have been considered by Wahba (1981) and Alfed et al (1996). A frequently used technique is that of series estimators based on spherical harmonics [see Abrial et al (2008) for example], which -roughly speaking -generalise estimators of a regression function on the line based on Fourier series to data on the sphere.…”